A Classification Approach for Open Manifolds

For closed manifolds there exists an effective highly elaborated classification approach the main steps of which are the Thom–Pontrjagin construction, bordism theory, surgery, Wall groups, the exact sequence of Browder–Novikov–Sullivan–Wall. All this can be expressed in an algebraic language of f. g. Zπ–, unitary Zπ–, Qπ–modules and their K–theory etc. Moreover, there exist many number valued invariants like classical characteristic numbers, signature, higher signature, analytic and Reidemeister torsion, η–invariants. For open manifolds, absolutely nothing of this remains, at least at the first glance. We have the following simple